Tangent Theorem 1

In this post, we discuss a theorem on tangent of circles.  We show that if a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. In the figure below, line m is perpendicular to OA at A. We show that it is a tangent to circle O.

Theorem

If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.

Given

Line m is perpendicular to OA at A.

Prove

Line m is tangent to circle O 

Proof

Let B be another point on line m other than A.

tangent theorem

Since OA is perpendicular m, triangle OAB is a right triangle with hypotenuse OB. This means that OB > AB and B must be on the exterior of the circle. Therefore B cannot lie on the circle and B is the only point of m that is on the circle. So, m is a tangent to the circle.

The converse of this statement is also true (see proof).

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